Summary

General Principles

The electric potential V, like the electric field, is created by source charges. Two major tools for calculating the potential are:

Use superposition: $V = V_{1} + V_{2} + V_{3} + \dots$

Model Model as a simple charge distribution.

Visualize Draw a pictorial representation.

Solve Set up a sum.

Divide the charge into point-like $Δ Q$ .
Find the potential due to each $Δ Q$ .
Use the charge density $(λ or η)$ to replace $Δ Q$ with an integration coordinate, then sum by integrating.

V is easier to calculate than $\vec{E}$ because potential is a scalar.

If charge q is placed in an electric potential V, the system’s electric potential energy (interaction energy) is

U_{elec} = q V

The electric potential energy of two point charges is

U_{q_{1} + q_{2}} = \frac{K q_{1} q_{2}}{r} = \frac{1}{4 π ϵ_{0}} \frac{q_{1} q_{2}}{r}

The potential energy of two opposite charges is negative.

The potential energy in an electric field of an electric dipole with dipole moment $\vec{p}$ is

U_{elec} = - p E cos θ = - \vec{p} \cdot \vec{E}

Model Model as an isolated system.

Visualize Draw a before-and-after representation.

Solve Mechanical energy is conserved.

Graphical representations of the potential:

Same as a point charge if $r \geq R$

$V = E s$ , where s is measured from the negative plate. The electric field inside is

E = \frac{Δ V_{C}}{d}

Electric potential: $1 V = 1 J / C$

Electric field: $1 V/m = 1 N / C$